W6D11: Feb. 8th, 2022

Functions of Random Varibales and Their Distributions

Ex. Uniform Distribution X, Let W = 2x+4

CDF of W

PDF of W

\(T=\frac{x^2}{4}+1\)

CDF and PDF of T

W6D12: Feb. 10th, 2022

Ex. R=ln(X)

CDF and PDF of R

Univariate Transformations Using CDF’s

Uniform X and \(T=\frac{x^2}{4}+1\)

Increasing vs. Decreasing functions

Method of Transformations

Ex. Find the pdf of \(Y=e^{x}\) and \(Z=\frac{1}{X}\)

### Complete pdf of Z

W7D13: Feb. 15th, 2022

Find the pdf of \(Y=3X\)

Find the pdf of \(Y=X^2\)

Bivariate Transformations using Jacobians

Ex. Total Volume Shipped

Ex. Revenue from sales

Ex. Difference in efficiency between economy and name-brand gas

Ex. Uniform Distribution

Bivariate Transfomations for Continuous rv using Jacobian

Find joint pdf of \(U_1\) and $U_2

W7D14: Feb. 17th, 2022

Joint PDF of \(U_1\) and \(U_2\) and PDF of \(U_1\)

Exponential \(Y_1\) and \(Y_2\), Let \(U_1=\frac{Y_1}{Y_2}\)

The joint pdf of \(U_1\) and \(U_2\)

Marginal PDF of \(U_1\)

W8D15: Feb. 22nd, 2022

6.5 The Method of Moment-Generating Functions

Distributions, mgf, pmf or pdf

Theorem 1

Example 3

Example 4

Theorem 6.2: MGF of sum of independent rv

Example

W8D16: Feb. 24th, 2022

Theorem 6.3

Example 6

Sampling Distributions and the Central Limit Theorem

Random Sample

W9D17: Mar. 1st, 2022

Example 1

Definition 7.1: Statistic

Definition: Sampling Distribution

Example 2

7.2 Sampling from the Normal Distribution

W9D18: Mar. 3rd, 2022

Example 3

Example 4

Theorem 7.3

Definition 4.10 and Theorem 7.2

TI Caclulator instructions and Example 5

W10D19: Mar. 8th, 2022

Inference

Derieved Distributions

Student’s T Distribution

Example 6

Central Limit Theorem

Example 7

W10D20: Mar. 10th, 2022

Final

  • will cover chapter 6 and 7

  • one sheet of notes (front and back)

  • integration calculator

  • mgf table

  • read homework feedback

  • practice mgf problems from the homework

  • write reason on steps for full points (i.e. is independent)

  • Remote exam join 30 minutes early

(None of the following will be on the final)

Methods

  • CDF

  • Jacobian (univariate)

  • MGF

  • Bivariate Jacobian

(Nothing below this point will be on the final)

Ch. 8 Statistical Ingerence

Introduction

Statistical Ingerence

Point Estimation

Unbiased and Biased Estimators

Definition Bias

Sample Mean

Mean Square Error (MSE)

Common Unbiased Point Estimators

W11D21: Mar. 15th, 2022

FINAL!!!